As an example we consider the 2D discrete Laplacian on a regular grid:

That is, they are the only positive functions with eigenvalue 1 for the discrete Laplacian ( sum of adjacent vertices minus value of vertex ) – the positive solutions to the homogeneous equation:

The Ramanujan – Petersson conjecture for general linear groups implies Selberg's conjecture about eigenvalues of the Laplacian for some discrete groups.

Finally there is another link that should be mentioned: Kirchhoff's theorem relates the number of spanning trees of a graph G to the eigenvalues of the discrete Laplacian.

* Richard Kenyon, The asymptotic determinant of the discrete Laplacian, Acta Math.

Its simplest combinatorial version is due to Zuk: let be a discrete group generated by a finite subset S, closed under taking inverses and not containing the identity, and define a finite graph with vertices S and an edge between g and h whenever g < sup >− 1 </ sup > h lies in S. If this graph is connected and the smallest non-zero eigenvalue of its Laplacian is greater than ½, then has property ( T ).

For the case of a finite-dimensional graph ( having a finite number of edges and vertices ), the discrete Laplace operator is more commonly called the Laplacian matrix.

There are various definitions of the discrete Laplacian for graphs, differing by sign and scale factor ( sometimes one averages over the neighboring vertices, other times one just sums ; this makes no difference for a regular graph ).

Then, the discrete Laplacian acting on is defined by

Closely related to the discrete Laplacian is the averaging operator:

If the grid size h = 1, the result is the negative discrete Laplacian on the graph, which is the square lattice grid.

For one, two and three dimensional signals, the discrete Laplacian can be given as convolution with the following kernels:

The spectrum of the discrete Laplacian is of key interest ; since it is a self-adjoint operator, it has a real spectrum.

Certain equations involving the discrete Laplacian only have solutions on the simply-laced Dynkin diagrams ( all edges multiplicity 1 ), and are an example of the ADE classification.

* Layered networks, the discrete Laplacian, and a continued fraction identity, Owen D. Biesel, David V. Ingerman, James A. Morrow, and William T. Shore

Cheeger's inequality from Riemannian Geometry has a discrete analogue involving the Laplacian Matrix ; this is perhaps the most important theorem in Spectral Graph theory and one of the most useful facts in algorithmic applications.

In many cases it is not possible to divide the process into a finite number of discrete stages, since the state of the stream is transformed in a continuous manner through the process.

This is the most common conception, and it attempts to describe a task in discrete, " mechanical " means.

Computers ( and computors ), models of computation: A computer ( or human " computor ") is a restricted type of machine, a " discrete deterministic mechanical device " that blindly follows its instructions.

The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny quantities has been around for millennia, but these ideas were founded in abstract, philosophical reasoning rather than experimentation and empirical observation.

It differs from a digital signal, in which a continuous quantity is represented by a discrete function which can only take on one of a finite number of values.

For an example of its use, analysis of the concentration of elements is important in managing a nuclear reactor, so nuclear scientists will analyze neutron activation to develop discrete measurements within vast samples.

Analytic geometry is widely used in physics and engineering, and is the foundation of most modern fields of geometry, including algebraic, differential, discrete, and computational geometry.

That is, the discrete molecular nature of a gas is ignored.

The discrete equivalent of the notion of antiderivative is antidifference.

The same result is true if the discrete point masses discussed above are replaced by a continuous distribution of matter.

Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures.

For example, an 8-bit CPU deals with a range of numbers that can be represented by eight binary digits ( each digit having two possible values ), that is, 2 < sup > 8 </ sup > or 256 discrete numbers.

It is a form of intellectual property ( like the patent, the trademark, and the trade secret ) applicable to any expressible form of an idea or information that is substantive and discrete.

These sheets are rolled at specific and discrete (" chiral ") angles, and the combination of the rolling angle and radius decides the nanotube properties ; for example, whether the individual nanotube shell is a metal or semiconductor.

The conspiracy is held to be responsible for a limited, discrete event or set of events.

Costume jewelry is considered a discrete category of fashion accessory, and displays many characteristics of a self-contained industry.

The work factor for breaking Diffie-Hellman is based on the discrete logarithm problem, which is related to the integer factorization problem on which RSA's strength is based.

The Copenhagen Interpretation denies that the wave function is anything more than a theoretical concept, or is at least non-committal about its being a discrete entity or a discernible component of some discrete entity.

In particular, the circular convolution can be defined for periodic functions ( that is, functions on the circle ), and the discrete convolution can be defined for functions on the set of integers.

Conversely, when one wants to compute an arbitrary number ( N ) of discrete samples of one cycle of a continuous DTFT, it can be done by computing the relatively simple DFT of s < sub > N </ sub >, as defined above.

* Pro-finite groups are defined as inverse limits of ( discrete ) finite groups.

The CDF of a discrete distribution will consist mostly of flat areas along with sudden jumps at each outcome defined in the sample space, while the CDF of a continuous distribution will typically rise gradually and continuously.

The history of the growth, decline and resurgence of rail transport can be divided up into several discrete periods defined by the principal means of motive power used.

Discrete time signal processing is for sampled signals that are considered as defined only at discrete points in time, and as such are quantized in time, but not in magnitude.

Its discrete analog is the Kronecker delta function which is usually defined on a finite domain and takes values 0 and 1.

A discrete function could be defined explicitly by a list, or by a formula for f ( n ) or it could be given implicitly by a recurrence relation or difference equation.

The discrete aurorae are sharply defined features within the diffuse aurora which vary in brightness from just barely visible to the naked eye to bright enough to read a newspaper at night.

* the discrete topology on X is defined by letting every subset of X be open ( and hence also closed ), and X is a discrete topological space if it is equipped with its discrete topology ;

* the discrete uniformity on X is defined by letting every superset of the diagonal

Terminal objects in a category C may also be defined as limits of the unique empty diagram ∅ → C. Since the empty category is vacuously a discrete category, a terminal object can be thought of as an empty product ( a product is indeed the limit of the discrete diagram

Usually a Markov chain is defined for a discrete set of times ( i. e., a discrete-time Markov chain ) although some authors use the same terminology where " time " can take continuous values.

This sum is defined only if the index is discrete.

Usually a Markov chain would be defined for a discrete set of times ( i. e. a discrete-time Markov Chain ) although some authors use the same terminology where " time " can take continuous values.

where is the discrete filter and is the discrete-time Fourier transform defined on the specified set of coordinates.

An n-fold categorical product can be defined as the limit with respect to a diagram given by the discrete category with n objects.

As the offspring of world and pop music, these categories are discrete from what is classically defined as world music, though most often, such hybrid genres are only given the choice of world with which to catalog themselves.

Functions related to the hardness of the discrete logarithm problem ( either modulo a prime or in a group defined over an elliptic curve ) are not known to be trapdoor functions, because there is no known " trapdoor " information about the group that enables the efficient computation of discrete logs.

Kendell and Jablinsky ( 2003 ) emphasized the importance of distinguishing between validity and utility, and argued that diagnostic categories defined by their syndromes should be regarded as valid only if they have been shown to be discrete entities with natural boundaries that separate them from other disorders.